Simplify the following expression and state the condition under which the simplification is valid. You can assume that $k \neq 0$. $p = \dfrac{10(k + 7)}{-3} \div \dfrac{4k(k + 7)}{4k} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{10(k + 7)}{-3} \times \dfrac{4k}{4k(k + 7)} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ 10(k + 7) \times 4k } { -3 \times 4k(k + 7) } $ $ p = \dfrac{40k(k + 7)}{-12k(k + 7)} $ We can cancel the $k + 7$ so long as $k + 7 \neq 0$ Therefore $k \neq -7$ $p = \dfrac{40k \cancel{(k + 7})}{-12k \cancel{(k + 7)}} = -\dfrac{40k}{12k} = -\dfrac{10}{3} $